Sample and population variances: elementary question

Given a sample of a normally distributed population, then the sample variance ≈the population variance divided by the sample size. Nice. However, if one now increases the sample size to the population, this becomes that the population variance ≈ the population variance divided by the population size, which is absurd. What elementary concept am I missing here? Thanks in advance

Given a sample of a normally distributed population, then the sample variance ≈the population variance divided by the sample size. Nice. However, if one now increases the sample size to the population, this becomes that the population variance ≈ the population variance divided by the population size, which is absurd. What elementary concept am I missing here? Thanks in advance

In statistics the idea is that you have a population and are trying to figure out its parameters. So you take a sample to get an estimate.

The population parameters are assumed to be constants. The sample parameters are random variables, because they will vary from sample to sample.

You are also confusing the sample variance with the variance of the sample mean.

The variance of the sample mean (usually) converges to zero, while of course the population variance does not. The sample variance converges to the population variance.

This stuff is confusing, but it is important to get it straight or you will never understand statistics. So good for you for asking.

Sample variance is NOT equal to population variance divided by sample size.

Yes, I know, that was the absurdity in my mini-proof that something was wrong with the original assumptions. That is, if I make a point that X is wrong because it leads to 1=0, then saying that 1≠0 is missing the point.